It’s going to depend on where she was shot in the arm, if she’s used to dealing with shock, and if any arteries were nicked. If we’re talking about your average middle class sixteen year old (male or female) who has never been around guns and never been in any physical danger then it’s unlikely. If we’re talking about a sixteen year old who grew up around guns and in situations where physical violence was a part of their general experience growing up, then it’s possible.
There are still some problems, however.
1) The person who shot her probably still has the gun and they’ve already proven that they’re willing to kill her. (They proved that the moment they drew the gun, firing it was just extra incentive.)
2) How is she going to knock them out? When someone gets shot while standing, the shock of it of it tends to make them automatically sink to the floor. Curling up into a ball, clutching the arm, and tucking in the knees, is a natural safety response that the body practices when it’s injured. If she’s not trained to fight that response, she probably won’t. So, the person who shot her is still standing and she’s now sitting. If she’s going to knock them out, then she’ll need some kind of ranged weapon that she can use while on the floor. This is preferably one that she brought with her or that got knocked away in the earlier scuffle. Something like a taser would be a good idea. The character, however, has to be willing to risk death (has to believe the person who shot them wants to kill them) and that there’s no way out except to fight back. This is not most people, especially not those who have grown up in a privileged environment.
3) Fighting makes the body bleed out faster. No matter where she was shot, she’s still bleeding. Shock, fear, and then combat all elevate your heart rate and the blood starts pumping through the body faster. If there is a hole in the body, then it causes the injured person to bleed out faster. This is why you’re supposed to sit down and breathe slowly until help arrives. Your character’s body has a limited amount of blood and the moment she got shot, the clock started ticking. She may not realize it, but if she gets up and fights, she’ll only be losing blood faster. She’s going to be at odds with her body’s own instinct to survive if she’s going to go after the person who shot her, because at this point fight or flight is dead.
The most important question here?
Is your sixteen year old willing to die to knock out (or kill) the person who harmed her?
If the answer is yes, then yes, she can fight back. If it’s no, then she probably can’t.